SC Model: Electrostatic Pressure

REVIEW AND COMPARISON OF THREE THIN FILM INSTABILITY MODELS

2.3 Linear Stability Analysis

2.3.1 SC Model: Electrostatic Pressure

Within the SC model the driving force is posited to be electrostatic in origin. In the work of Chou and Zhuang [2, 3], there was assumed to be a surface charge density along the interface which would induce image charges in the upper and lower bounding plates which were grounded, as illustrated in Fig. 2.2. The presence of the electric charges creates an electric field which they hypothesized was responsible for the deformation of the interface. Because the AP model had not yet been published by Schäffer et al., the net pressure from acoustic phonon reflections is zero and so P_ac = 0. Furthermore, they did not consider the surface tension to vary with any external field which implies that ∇f_sΓ = 0. All that remains is to define the electric pressure, Pel, created by the interfacial charge density and complete the linear stability analysis.

The electrostatic pressure arises from the difference between the Maxwell stress tensors, T^em_i , in the air and nanofilm layers. Explicitly, the magnitude of the pressure in the normal direction is

Pel = 1 Pc

nˆ· T^em_air −T^em_film

·n.ˆ (2.48)

The Maxwell stress tensor in matter without any magnetic fields has the form [20]

T^em = E®D® − 1 2I

E®· ®D

, (2.49)

where E® is the electric field and D® = εoεE® is the electric displacement field. εo

is the the permittivity of free space. Note that ε is the relative permittivity of the medium, and is distinct from which is the long-wavelength expansion parameter.

In air we assume that the relative permittivity is equal to unity, so thatεair = 1. To proceed further, the electric fields in both the air and film layers are solved using Laplace’s equation and then the Maxwell stress tensors are computed. These are

then inserted into the electrostatic pressure term,P_el. OnceP_elhas been computed, linear stability analysis is applied to the resulting thin film height evolution equation to find the wavevector and growth rate of the fastest growing mode.

Electrostatic Governing Equations

Within the derivation of Chou and Zhuang, it was assumed that there are no signif- icant magnetic fields present in the system. This reduces the problem of solving for the electric field within the system to a simple electrostatics problem. Furthermore, it was assumed that there was no volumetric charge density present within either the air or film layers and that the only charge is present at the interface between the two layers. The interfacial charge density is constant during deformation and denoted byσ. These assumptions imply that the governing differential equation is Laplace’s equation

∇²φi = 0. (2.50)

In this expression φi is the potential in theith layer. Since there is no externally applied voltage in this system, both the upper and lower bounding plates are assumed to be grounded so that

φ_film(z= 0)=0, (2.51)

φair(z= d)=0. (2.52)

Along the interface, the usual electrostatic boundary conditions are applied [20]

ˆ n·

D®_air− ®D_film

= εonˆ·

E®_air−ε_filmE®_film

= σ, (2.53)

ˆ n×

E®_air− ®E_film

= 0. (2.54)

Finally, the relationship between the electric field and the electric potential is

E®i =−∇φi. (2.55)

Scaled Electrostatic Equations

To scale the electrostatic equations, the same scalings which were defined in Sec. 2.2 are used but there are two more for the electric potential and the electric field.

φei = φi

Φ_c;eE®i = E®iho

Φ_c . (2.56)

The quantityΦ_cis a characteristic potential which will be determined in the course of scaling the equations, similar to howPcandΓ_cwere determined above. The rela- tionship between the nondimensional electric potential and electric field transforms from Eq. (2.55) to

e®

Ei= −∇f_kφei− ∂φei

∂Z. (2.57)

Once we nondimensionalize Laplace’s equation from Eq. (2.50) we find that to second order

∂φei

∂Z² = 0. (2.58)

The exterior Dirichlet boundary conditions simply become

φe_film(Z = 0)=0, (2.59)

φeair(Z = D)=0. (2.60)

The tangential electrostatic boundary condition of Eq. (2.54) is equivalent to the requirement that the potential be continuous across the interface. Therefore,

φe_film(Z = H)= φe_air(Z = H). (2.61) The final electrostatic boundary equation is the one shown in Eq. (2.53) for the normal components of the electric displacement field at the interface. Using the scaled normal vector which was derived above in Eq. (2.24), this yields

Φcεo

ho

−∇f_kH+Zˆ

·

−∇f_kφeair−εo

∂φeair

∂Z Zˆ +εfilm∇f_kφefilm+εfilm

∂φefilm

∂Z Zˆ

= σ.

From this it is clear that all the tangential terms in this equation are order²and can be neglected. Furthermore, the characteristic electric potential scale arises from the charge density at the interface and should be

Φ_c = σho

εo

. (2.62)

This boundary condition then simplifies to ε_film∂φe_film(Z = H)

∂Z − ∂φe_air(Z = H)

∂Z = 1. (2.63)

Electric Field Solution

The scaled Laplace equation from Eq. (2.58) was integrated twice with respect to Z, yielding electric potentials in each layer that are linear.

φe_film = A^SC_filmZ +B^SC_film, φe_air= A^SC_airZ +B_air^SC.

In this equation A^SC_film, B^SC_film, A^SC_air, and B_air^SC are integration constants. The Dirichlet boundary conditions on the bounding plates from Eqs. (2.59) and (2.60) imply that B_film^SC = 0 andB^SC_air = −D A^SC_air

φefilm = A^SC_filmZ, φe_air = A^SC_air(Z−D).

The electric potential must be continuous across Z = Haccording to the boundary condition in Eq. (2.61), so that A^SC_filmcan be expressed in terms of A^SC_air

A^SC_film = A^SC_air(H−D) H .

This implies that the electric potentials should have the form φe_film = A^SC_airZ(H−D)

H , φe_air = A^SC_air(Z−D).

The only remaining boundary condition is Eq. (2.63) and this implies that the one remaining integration constant is

A^SC_air = H

(ε_film−1)H−ε_filmD. Returning to the electric potentials, they have the form

φefilm = Z(H−D)

(ε_film−1)H−ε_filmD, (2.64) φeair = H(Z−D)

(ε_film−1)H−ε_filmD. (2.65)

Based on the relation in Eq. (2.57) between the electric potential and the electric field, the nondimensional electric fields at the interface are

e®

E_film= DH∇f_kH

[ε_filmD− (ε_film−1)H]²

!

− (D−H) εfilmD− (εfilm−1)H

Z,ˆ (2.66)

e®

E_air= ε_filmD(H−D)f∇_kH [ε_filmD− (ε_film−1)H]²

!

− H

εfilmD− (εfilm−1)H

Z.ˆ (2.67)

The most important thing to note about these electric fields is that the vertical components do not have an, while the horizontal components are first order in. This means that when these electric fields are inserted into the Maxwell stress tensor, all terms which contain products with two tangential components, such as ExEy, ExEx, andEyEy, are order²and can be neglected. Computing the expression for the normal component of the stress tensor dotted into the normal vector yields

ˆ

n·T^em·nˆ =εoε

−∂H

∂X −∂H

∂Y 1























−1

2E_z² 0 ExEz

0 −1

2E_z² EyEz

ExEz E_yEz

1 2E_z²













































−∂H

∂X

−∂H

∂Y 1























= εoε 2 E_z².

Recalling the form of the electric pressure from Eq. (2.48), the electric pressure is P_el = εo

2Pc

E_air,z² −ε_filmE_film,z² .

In terms of the electric fields which are expressed in Eqs. (2.66) and (2.67), this pressure becomes

P_el = σ² 2εoPc

(1−ε_film)H²+2ε_filmDH−ε_filmD² [εfilmD− (εfilm−1)H]²

. (2.68)

Linear Stability Predictions

Returning to the height evolution equation in Eq. (2.44), the gradient of the elec- trostatic pressure was computed and substituted yielding the following expression

∂H

∂τ +∇f_k · H³

3Ca

∇f³_kH

+ H³σ² 3εoPc

ε_filmD²

[ε_filmD− (ε_film−1)H]³

∇f_kH

=0. (2.69)

Insertion of the linear stability perturbation function from Eq. (2.45) and cancellation of the common exponentials yields a nondimensional dispersion relation where terms of orderfδh²have been dropped

β^SC(K)+ K⁴ 3Ca

− σ² 3εoPc

εfilmD²

[ε_filmD− (ε_film−1)]³

K² =0. (2.70) This specific dispersion relation has a representative form that will be borne out in the other proposed models. The dispersion relations for each model are of the general form

β(K)= A2K²− A4K⁴, (2.71) where A₂and A₄ are constants whose exact form depends on the model. As such, the location and magnitude of the maximum growth rate can be found from this general form. The mode with the maximum growth rate is assumed to be the one observed experimentally so the wavevector at which this maximum occurs should then correspond to the characteristic wavelength of the real space pattern which is observed. The form of the dispersion relation in Eq. (2.71) can be solved analytically for the wavevector corresponding to the maximum growth rate. This maximum wavevector is denoted byKo

Ko= r A₂

2A₄. (2.72)

The maximum value of the growth rate is then βo≡ β(Ko)= A²₂

4A₄. (2.73)

For the SC model, A₂and A₄are A^SC₂ = σ²

3εoPc

ε_filmD²

[ε_filmD− (ε_film−1)]³

, (2.74)

A^SC₄ = 1

3Ca. (2.75)

Consequently,Koandβofor the SC model are K_o^SC=

s σ²hoD² 2εoε²_filmγ²

D+ 1 ε_film −1

⁻3/2

, (2.76)

β_o^SC= λoho

3µucγ

σ²D² 2εoε²_film

!2

D+ 1 ε_film −1

−6

. (2.77)

Figure 2.3: Instability geometry in AP model

λ_o Air

Molten Nanofilm ho

h(x, t)

do ∆T

T_H T_C

z x

×y

The distinguishing feature of the AP model is the coherent propagation of acoustic phonons through the bilayers, which create a destabilizing radiation pressure.

These are the same quantities as those derived by Zhuang [3], just expressed in nondimensional terms. The dimensional quantities will be presented in Sec. 2.3.4 with the results from the other two models.